find k such that for every real x we have (1 + kx)/(1 + x^2) < k

Assuming k is any real number... 
 
leads to
 
then 
 
and combining them we get 
 
after simplifying the numerator... 
 
Notice that  is always positive, so to determine when the sign of the expression will be less than zero we only need to look at the numerator: 
 
using the quadratic formula to find x... 
 
then... 
 
then... 
 
which only has a positive real value when 
 and  

or 
 and  
So we notice x is only real on the interval 
 
This interval gives us boundary conditions to test. After testing (using a graphing utility) we see only the values k>4/3 satisfies the inequality for ALL values of x